So far, the evidence suggests that there is such a fixed point, known as the Reuter fixed point [1]}, [2]}, [3]}, [4]}, [5]}, and that it possesses three relevant couplings in pure gravity [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}. However, not all couplings need to reach a fixed point for the theory to be asymptotically safe, since one has the freedom to perform field reparameterisations which can be used to eliminate the so-called inessential couplings from the RG equations [18]}.
The inessential couplings do not appear in expressions for observables, such as cross sections and reaction rates, and, therefore, can take different values without affecting the physics.
Couplings, therefore, fall into two classes: the essential couplings \(\lambda _a\)  which enter into expressions for observables and the inessential couplings \(\zeta _\alpha \)  which are scheme dependent and unphysical.
Consequently, the scaling behaviour of inessential couplings is entirely scheme dependent and they must not be included in the set of relevant couplings [19]}. It follows that a coupling that may appear relevant could turn out to be inessential and, therefore, does not contribute to the counting of free parameters.
Although the potential existence of inessential couplings has been pointed out [20]}, [21]}, [22]}, [23]}, [24]}, they have been almost universally ignored in investigations of asymptotic safety. In particular, attempts to find a suitable fixed point have required fixed points for all gravitational couplings, included in a given approximation, instead of incorporating field reparameterisations into the RG equations and checking which of the couplings are inessential. Here we shall remedy this oversight by incorporating field reparameterisations in the gravitational RG equations which allow us to eliminate the inessential couplings from the flow equations. To do so we will utilise the essential RG approach, which has been put forward in [25]}, where we only compute the flow of the essential couplings.
